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SUMMARY:From AdS to BEC (dynamics in spatially confined Hamiltonian system
 s) - Piotr Bizon\, Jagiellonian University
DTSTART:20180307T141500Z
DTEND:20180307T151500Z
UID:TALK97261@talks.cam.ac.uk
CONTACT:David Marsh
DESCRIPTION: The long-time behavior of nonlinear dispersive waves subject 
 to spatial confinement can be very rich and complex because\, in contrast 
 to unbounded domains\, waves cannot escape to infinity and keep self-inter
 acting for all times. If\, in addition\, the linear spectrum around the gr
 ound state is fully resonant\, then the nonlinearity can produce significa
 nt effects for arbitrarily small perturbations. The weak field dynamics of
  such systems can be approximated by solutions of the corresponding infini
 te-dimensional time-averaged Hamiltonian systems which govern resonant int
 eractions between the modes. A major mathematical challenge in this contex
 t is to describe the energy transfer between the modes. I will discuss thi
 s problem for several spatially confined systems: a cubic wave equation on
  the 3-sphere\, the Einstein-scalar system  with negative cosmological con
 stant (modeling the weakly turbulent behavior of small perturbations of th
 e anti-de Sitter spacetime)\, the nonlinear Schroedinger equation with a t
 rapping potential (modeling  the dynamics of a Bose-Einstein condensate)\,
  and the Schroedinger-Newton-Hooke system describing a harmonically trappe
 d self-gravitating condensate.  Some intriguing parallels between these sy
 stems will be emphasized.
LOCATION:MR2\, Centre for Mathematical Sciences
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